## Iterative algorithms for hierarchical fixed points problems and variational inequalities.(English)Zbl 1205.65192

Summary: This paper deals with a method for approximating a solution of the fixed point problem: find $$\tilde{x}\in H$$; $$\tilde{x}=(\mathrm{proj}_{F(t)}S)\tilde{x}$$, where $$H$$ is a Hilbert space, $$S$$ is some nonlinear operator and $$T$$ is a nonexpansive mapping on a closed convex subset $$C$$ and $$\mathrm{proj}_{F(t)}$$ denotes the metric projection on the set of fixed points of $$T$$. First, we prove a strong convergence theorem by using a projection method which solves some variational inequality. As a special case, this projection method also solves some minimization problems. Secondly, under different restrictions on parameters, we obtain another strong convergence result which solves the above fixed point problem.

### MSC:

 65J15 Numerical solutions to equations with nonlinear operators 65K15 Numerical methods for variational inequalities and related problems 49J40 Variational inequalities
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### References:

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